Closed-Form Stiffness Matrix for the Four-Node Quadrilateral Element with a Fully Populated Material Stiffness

This technical note presents closed-form finite-element stiffness formulations for the four-node quadrilateral element with a fully populated material stiffness, which is required for the nonlinear analysis of reinforced concrete membrane structures. With the material stiffness matrix accounting for anisotropy of the materials and prestrain effects, the developed closed-form element stiffness can be incorporated into a nonlinear finite-element algorithm. Through use of the developed explicit expressions, the examples provided show that the computational effort required to form the stiffness matrix is greatly reduced, compared to either the conventional numerical integration scheme or the elastic-material-stiffness-oriented Griffths’ FORTRAN subroutine.